Zero Matrix is Identity for Matrix Entrywise Addition over Ring
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Theorem
Let $\struct {R, +, \circ}$ be a ring.
Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$.
Let $\mathbf 0_R = \sqbrk {0_R}_{m n}$ be the zero matrix of $\map {\MM_R} {m, n}$.
Then $\mathbf 0_R$ is the identity element for matrix entrywise addition.
Proof 1
Let $\mathbf A = \sqbrk a_{m n} \in \map {\MM_R} {m, n}$.
Then:
\(\ds \mathbf A + \mathbf 0_R\) | \(=\) | \(\ds \sqbrk a_{m n} + \sqbrk {0_R}_{m n}\) | Definition of $\mathbf A$ and $\mathbf 0_R$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqbrk {a + 0_R}_{m n}\) | Definition of Matrix Entrywise Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqbrk a_{m n}\) | Ring Axiom $\text A3$: Identity for Addition is $0_R$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \mathbf A + \mathbf 0_R\) | \(=\) | \(\ds \mathbf A\) | Definition of Zero Matrix over Ring |
Similarly:
\(\ds \mathbf 0_R + \mathbf A\) | \(=\) | \(\ds \sqbrk {0_R}_{m n} + \sqbrk a_{m n}\) | Definition of $\mathbf A$ and $\mathbf 0_R$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqbrk {0_R + a}_{m n}\) | Definition of Matrix Entrywise Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqbrk a_{m n}\) | Ring Axiom $\text A3$: Identity for Addition is $0_R$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \mathbf 0_R + \mathbf A\) | \(=\) | \(\ds \mathbf A\) | Definition of Zero Matrix over Ring |
$\blacksquare$
Proof 2
By definition, matrix entrywise addition is the Hadamard product with respect to ring addition.
We have from Ring Axiom $\text A3$: Identity for Addition that the identity element of ring addition is the ring zero $0_R$.
The result then follows directly from Zero Matrix is Identity for Hadamard Product.
$\blacksquare$
Also see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices